Balance in Bayesian Analyses

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# Balance in Bayesian Analyses
## Dr. Mine Dogucu
Examples from [bayesrulesbook.com](https://bayesrulesbook.com)

]

---

__Bechdel Test__

Alison Bechdel’s 1985 comic Dykes to Watch Out For has a strip called [The Rule](https://www.npr.org/templates/story/story.php?storyId=94202522?storyId=94202522) where a person states that they only go to a movie if it satisfies the
following three rules:
- the movie has to have at least two women in it;
- these two women talk to each other; and
- they talk about something besides a man.

This test is used for assessing movies in terms of representation of women. Even though there are three criteria, a movie either fails or passes the Bechdel test.

---

### Different Priors, Different Posteriors

Let `\(\pi\)` be the the proportion of movies that pass the Bechdel test.

Below there are three different people with three different priors about `\(\pi\)`.

<table align = "center">
  <tr>
    <td>optimist</td>
    <td>clueless </td>
    <td>feminist </td>
  </tr>
  
  <tr>
    <td>Beta(14,1)</td>
    <td>Beta(1,1)</td>
    <td>Beta(5,11)</td>
  </tr>

</table>

Plot their priors.

---

## Priors

---

### Vocabulary

__Informative prior:__ An informative prior reflects specific information about the unknown
variable with high certainty (ie. low variability).

__Vague (diffuse) prior:__

A vague or diffuse prior reflects little specific information about the unknown variable. A flat prior, which assigns equal prior plausibility to all possible values of the variable, is a special case.

---

## Data

- `library(bayesrules)` has `bechdel` data frame. Randomly select 20 movies from this dataset (seed = 84735)

- Based on observed data, update the posterior for all three people.

- Calculate the summary statistics for the prior and the posterior for all three.

- Plot the prior, likelihood, and the posterior for all three.

- Explain the effect of different priors on the posterior.

---

```r
library(tidyverse)
library(bayesrules)
set.seed(84735)
```

```r
bechdel_sample <- sample_n(bechdel, 20)
```

```r
count(bechdel_sample, binary)
```

```
## # A tibble: 2 x 2
##   binary     n
##   <chr>  <int>
## 1 FAIL      11
## 2 PASS       9
```

---

## The Optimist

```r
summarize_beta_binomial(14, 1, y = 9, n = 20)
```

```
##       model alpha beta      mean      mode         var         sd
## 1     prior    14    1 0.9333333 1.0000000 0.003888889 0.06236096
## 2 posterior    23   12 0.6571429 0.6666667 0.006258503 0.07911070
```

---

## The Optimist

```r
plot_beta_binomial(14, 1, y = 9, n = 20)
```

---

## The Clueless

```r
summarize_beta_binomial(1, 1, y = 9, n = 20)
```

```
##       model alpha beta      mean mode        var        sd
## 1     prior     1    1 0.5000000  NaN 0.08333333 0.2886751
## 2 posterior    10   12 0.4545455 0.45 0.01077973 0.1038255
```

---

## The Clueless

```r
plot_beta_binomial(1, 1, y = 9, n = 20)
```

---

## The Feminist

```r
summarize_beta_binomial(5, 11, y = 9, n = 20)
```

```
##       model alpha beta      mean      mode        var         sd
## 1     prior     5   11 0.3125000 0.2857143 0.01263787 0.11241827
## 2 posterior    14   22 0.3888889 0.3823529 0.00642309 0.08014418
```

---

## The Feminist

```r
plot_beta_binomial(5, 11, y = 9, n = 20)
```

---

## Comparison

---

### Different Data, Different Posteriors

Morteza, Nadide, and Ursula –  all share the optimistic Beta(14,1) prior for `\(\pi\)` but each have access to different data. Morteza reviews movies from 1991. Nadide reviews movies from 2000 and Ursula reviews movies from 2013. How will the posterior distribution for each differ?

---

## Morteza's analysis

```r
bechdel_1991 <- filter(bechdel, year == 1991)
count(bechdel_1991, binary)
```

```
## # A tibble: 2 x 2
##   binary     n
##   <chr>  <int>
## 1 FAIL       7
## 2 PASS       6
```

```r
6/13
```

```
## [1] 0.4615385
```

---

## Morteza's analysis

```r
plot_beta_binomial(14, 1, y = 6, n = 13)
```

---

## Nadide's analysis

```r
bechdel_2000 <- filter(bechdel, year == 2000)
count(bechdel_2000, binary)
```

```
## # A tibble: 2 x 2
##   binary     n
##   <chr>  <int>
## 1 FAIL      34
## 2 PASS      29
```

```r
29/(34+29)
```

```
## [1] 0.4603175
```

---

## Nadide's analysis

```r
plot_beta_binomial(14, 1, y = 29, n = 63)
```

---

## Ursula's analysis

```r
bechdel_2013 <- filter(bechdel, year == 2013)
count(bechdel_2013, binary)
```

```
## # A tibble: 2 x 2
##   binary     n
##   <chr>  <int>
## 1 FAIL      53
## 2 PASS      46
```

```r
46/(53+46)
```

```
## [1] 0.4646465
```

---

## Ursula's analysis

```r
plot_beta_binomial(14, 1, y = 46, n = 99)
```

---

## Summary